Is 0 even, odd, or neither? | Brilliant Math & Science Wiki (2024)

Is 0 even or odd?

Why some people say it's even: It's evenly divisible by 2.

Why some people say it's odd: It's not divisible by 2 and it's not a multiple of 2.

Why some people say it's both: Both of the top two arguments above are reasonable, so 0 is actually both even and odd!

Why some people say it's neither: Both of the arguments above are reasonable, but neither is completely true or sensible. Also, clearly, it can't be both.


The correct answer is that 0 is \( \color{green}{\textbf{even}}\), and not odd.

The definition of an even number:

Definition 1. A number is even if it is divisible by 2. (or "A number is even if it has 2 as a factor.")

Definition 2. A number is even if it is a multiple of 2.

Proof that 0 is even:

There are several common definitions of multiple and divisor, but all of them make \(0\) an even number.

The definition of a divisor (or factor):

\(D\) is a divisor of \(N\) if and only if \(\frac{N}{D}\) is an integer.

By this definition, \(0\) is even because \(\frac{0}{2} = 0,\) which is an integer.

An alternative definition of a divisor:

\(D\) is a divisor of \(N\) if and only if \(\frac{N}{D}\) has a remainder of \(0.\)

By this definition, \(0\) is even because \(\frac{0}{2} = 0,\) with a remainder of \(0.\)

The definition of a multiple:

An integer \(M\) is a multiple of an integer \(N\) if and only if there exists an integer, \(Z,\) such that \(N \times Z = M\).

By this definition, \(0\) is even because if we let \(Z\) be the integer \(0,\) then \(2 \times 0 = 0,\) therefore \(0\) is a multiple of \(2.\)

An interesting additional note is that, using the same logic, we can see that \(0\) is actually divisible by all integers other than itself \(\big(\)since \(\frac{0}{0}\) is undefined\(\big),\) and that \(0\) is a multiple of all integers.

The definition of an odd number:

Definition 1. "A number is odd if it is equal to \(2n + 1\) for some integer \(n.\)"

Common Notion: "A number is odd if it is an integer that is not even."
(Note: This common notion is true, but it's not considered the primary definition of "odd.")

Proof that 0 is not odd:

If \(2n + 1 = 0,\) then subtracting \(1\) from both sides, we see that \(2n = -1,\) and therefore \(n = -\frac{1}{2}.\) However, \(-\frac{1}{2}\) is not an integer, therefore \(0\) is not odd.

(This is a proof by contradiction.)


Rebuttal: \(2\) is not a factor of \(0\) because \(\frac{2}{0}\) is undefined.

Reply: You're mixing up the positions of the two variables in the definition of divisor when you set up that fraction. For example, by the same reasoning "\(5\) is not a factor of \(10\) because \(\frac{5}{10}\) is not an integer."

The correct definition of a factor is that \(D\) is a factor of \(N\) if and only if \(\frac{N}{D}\) is an integer. Notice that the potential divisor is the number in the denominator of the fraction. Therefore, the fraction that we set up to test if \(2\) is a factor of \(0\) is \(\frac{0}{2}.\) Since \(\frac{0}{2} = 0\) with remainder \(0,\) \(2\) is a factor of \(0\).

Therefore, what you can conclude from your claim that \(\frac{2}{0}\) is undefined is that \(0\) is not a factor of \(2.\) However, that does not pertain to the question of whether or not \(0\) is even.

Rebuttal: This is crazy. We need to come up with new definitions if the ones that we have imply that \(0\) has infinitely many factors and is a multiple of everything.

Reply: While, in the proofs above, we only thought about this issue from the logical perspective of verifying the accepted definition of a prime number, it's also important to realize that the definitions are worded as they are to create a system which is as sensible and usable as possible. Including 0 as part of every set of multiples is actually very natural. For example, consider the visual representations of multiples of \(N\) pictured below. It's clear that including 0 in the set of multiples for each number completes the pattern, whereas omitting 0 would create a strange exception/irregularity in each set.

Is 0 even, odd, or neither? | Brilliant Math & Science Wiki (1)

Acknowledging and preserving this kind of pattern creates symmetry in the mathematics and makes it more likely that the theorems and proofs which use these definitions can be simply stated, without many exceptions and special cases. For example, consider the theorem, "The sum of any two multiples of a number is also a multiple of that number." If \(0\) were not a multiple of every number, this elegant theorem would have to be revised to, "The sum of any two multiples of a number is either 0 or a multiple of that number, and the sum of 0 and any multiple of a number is also a multiple of that number."

This entire page is just a matter of definition. Mathematicians love to define things; they decide that \(0\) should be considered even because they can do so. But, of course, mathematicians also have reasons when defining things, and are not just making this decision at whim.

Want to make sure you've got this concept down? Try these problems:

Odd Even Both Neither

\[\large \text{Is } 0 \text{ even or odd?}\]

True False It's undefined

True or False?

\[{\color{blue}{0} \text{ is a multiple of } \color{red}{3}.}\]

0 is only divisible by itself 0 is only divisible by 1 0 is divisible by all integers besides 0 0 is divisible by all integers 0 is not divisible by any number

What are the integral divisors of 0?

\(\)
Clarification: Given two integers \(N\) and \(M,\) \(N\) is an "integral divisor" of \(M\) if \(F = \frac{M}{N}\) is an integer.

See Also

  • List of Common Misconceptions
  • Is 0 a prime number?
  • Is 0 a multiple of 3?
Is 0 even, odd, or neither? | Brilliant Math & Science Wiki (2024)

FAQs

Is 0 even, odd, or neither? | Brilliant Math & Science Wiki? ›

The correct answer is that 0 is even \color{#20A900}{\textbf{even}} even, and not odd. The definition of an even number: Definition 1. A number is even if it is divisible by 2.

Is 0 even or odd or neither? ›

Zero is an even number. In other words, its parity—the quality of aninteger being even or odd—is even. The simplest way to prove that zero iseven is to check that it fits the definition of "even": it is an integermultiple of 2, specifically 0 × 2.

Is 0 mathematically even? ›

In mathematics, zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": it is an integer multiple of 2, specifically 0 × 2.

Is 0 the first even number? ›

The first even whole numbers are: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and so on. Notice in the number line, that between 6 and 8, for example, there isn't any other even number. When this happens, the numbers are called consecutive even numbers. Similarly, 2 and 4, -6 and -4 are consecutive even numbers.

Why is 0 not the smallest even number? ›

Zero is considered an even number, and even though zero is necessary nothing, HOWEVER, there is NEGATIVE NUMBERS. -2 is smaller than 0, but as it is technically a 2, it is still an even number, and since negatives are lower than 0, -2 is a smaller even number. So no, 0 is not necessarily the smallest even number.

Why is zero not a number? ›

Zero could be considered a placeholder or a number. Zero is neither positive nor negative and thus it is considered a neutral number. Mathematicians agree zero is a counting number, a whole number, and an integer.

What type of number is 0? ›

Answer: 0 is a rational number, whole number, integer, and a real number. Let's analyze this in the following section. Explanation: Real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Does 0 exist in algebra? ›

Elementary algebra

The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number. All rational numbers are algebraic numbers, including 0. When the real numbers are extended to form the complex numbers, 0 becomes the origin of the complex plane.

Who invented zero? ›

Aryabhata, a great astronomer of the classic age of India was the one who invented the digit “0” (zero) for which he became immortal but later on is given to Brahmagupta who lived around a century later 22, another ancient Indian mathematician.

Would math exist without 0? ›

Having no zero would unleash utter chaos in the world. Maths would be different ball game altogether, with no fractions, no algebra and no calculus. A number line would go from -1 to 1 with nothing bridging the gap. Zero as a placeholder has lots of value and without it a billion would simply be “1”.

Is zero an integer or not? ›

The number zero is an integer. And It is the only integer which doesn't have any sign.

What is the only number that is both odd and even? ›

Every integer is either even or odd, and no integer is both even and odd. This includes 0, which is even.

Is 0 an irrational number? ›

The number 0 is present in rational numbers. The number 0 is not an irrational number. Real numbers: Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”.

Is 0 technically even? ›

Zero is an even number because it can be divided by two without producing a remainder. Learn how to prove 0 is even and why it isn't also an odd number.

How do you prove zero is even? ›

So, let's tackle 0 the same way as any other integer. When 0 is divided by 2, the resulting quotient turns out to also be 0—an integer, thereby classifying it as an even number.

What do mathematician call a number less than 0? ›

A negative number is any number that is less than zero.

Are numbers that end in 0 odd or even? ›

To identify even numbers, we observe the last digit or the ones digit of the number. If it ends in the digits 0, 2, 4, 6, or 8, then it is an even number. Otherwise, it is an odd number.

Does odd numbers start from 0? ›

The smallest odd number is 1 because odd numbers start from 1 and are listed as, 1, 3, 5, 7, 9, 11 and so on.

Which number is neither or even? ›

Expert-Verified Answer

all odd 1 digit numbers are: 1,3,5.7and 9 and all even 1 digit numbers are: 2, 4, 6, 8 and 0. hence all numbers (0 ,1 … 8, 9) are either odd or even numbers and so there is no counting number which is neither even nor odd.

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